Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rspcegf | Structured version Visualization version GIF version |
Description: A version of rspcev 3620 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
rspcegf.1 | ⊢ Ⅎ𝑥𝜓 |
rspcegf.2 | ⊢ Ⅎ𝑥𝐴 |
rspcegf.3 | ⊢ Ⅎ𝑥𝐵 |
rspcegf.4 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rspcegf | ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspcegf.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
2 | rspcegf.3 | . . . . . 6 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | nfel 2989 | . . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵 |
4 | rspcegf.1 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
5 | 3, 4 | nfan 1891 | . . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ 𝐵 ∧ 𝜓) |
6 | eleq1 2897 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵)) | |
7 | rspcegf.4 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
8 | 6, 7 | anbi12d 630 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑥 ∈ 𝐵 ∧ 𝜑) ↔ (𝐴 ∈ 𝐵 ∧ 𝜓))) |
9 | 1, 5, 8 | spcegf 3588 | . . 3 ⊢ (𝐴 ∈ 𝐵 → ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))) |
10 | 9 | anabsi5 665 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) |
11 | df-rex 3141 | . 2 ⊢ (∃𝑥 ∈ 𝐵 𝜑 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) | |
12 | 10, 11 | sylibr 235 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝜓) → ∃𝑥 ∈ 𝐵 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∃wex 1771 Ⅎwnf 1775 ∈ wcel 2105 Ⅎwnfc 2958 ∃wrex 3136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rex 3141 df-v 3494 |
This theorem is referenced by: rspcef 41211 stoweidlem46 42208 |
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